1. Field of the Invention
The present invention relates to a method of simulation and particularly relates to the method of simulating a target in a sputtering arrangement and also to the method of simulating a track of a sputtered particle in the arrangement.
2. Description of the Related Art
In the field of semiconductor integrated circuits, there has been a growing trend toward the design of high-density integrated circuits, as the technology of integrating electronic circuits progresses. This trend has brought about an urgent requirement to develop a technique for growing a semiconductor thin film in a finely formed contact hole of a highly integrated IC.
Optimization of a sputtering arrangement is imperative in order to meet the above requirement. Since the actual construction of and experimentation with the arrangement of the sputtering apparatus are problematic from the viewpoint of cost and development time, the optimization has been attempted by means of simulation.
In connection with the design of the sputtering arrangement, it has been required to improve the direction-dependent distribution characteristic of the particles ejected from the target, as well as to improve the shapes of components of the arrangement, such as a collimator. These improvements have been desired more anxiously, particularly as a high aspect ratio of a circuit element is required for fine-structuring of an IC. Hereinafter, the direction-dependent distribution of the particles ejected from the target is referred to as an angular distribution.
There have been presented methods of simulating the angular distribution using a method of the molecular dynamics (MD). Yamada H., the present inventor, has invented such a simulation method of sputtering, which method was disclosed in Japanese Patent Application. No. 55682/96 (filed by the present applicant) filed before the present application. Since the simulation method is relevant to the present invention, it will be referred to, hereinafter, as Yamada""s former method.
FIG. 1 is a flow chart to illustrate Yamada""s former method.
Referring to FIG. 1, the method includes steps P1, P2 and S5. In step P1, an angular distribution of the particles ejected from a target is calculated using the MD method and stored in a data file. In step P2, the angular distribution of ejection is successively read out. In step S5, a track of the sputtered particle in the sputter arrangement is calculated from the vertical and horizontal components of the direction of ejection by means of the Monte Carlo (MC) method, wherein the angular distribution of ejected particles read out in Step P2 is taken as an initial value.
In step P1, the velocity of the ion incident on the target is calculated from the applied voltage, and the surface temperature of the target is calculated by means of thermoanalysis. The initial velocity of the atoms in the interior of the target is derived from the surface temperature. The track of an interior atom of the target is calculated by means of the MD method on the basis of the derived initial velocity (step S11). The track of an atom farther than a cut-off distance from the surface of the target is next extracted as a sputtered atom (step S12), the horizontal and vertical components of the ejected directions of the sputtered atoms are extracted (step P13), and are stored in a data file (step S14). The steps S11-S14 are repeated until the number of data points reachs a predetermined number-N (step S15). The number of extracted sputtered atoms, i.e., the number of ejected angles of the particles N is of the order of 100-200 in order for the calculation to be carried out within a practical calculation time.
Next, the ejected direction is successively read out in step P2. In this step, the ejection angle data is read out successively from the data file starting with #01 data (step P21). It is judged then whether the number of the read-out data n reaches N (step P22). The read-out ejection angle data is successively served to the calculation of the tracks of sputtered atoms in the sputtering arrangement (step S51) until the number of data points read-out reaches N. Upon the number reaching N, #01 data is employed again for the calculation of the tracks (step P23), and the N data are repeatedly employed for the calculation of the tracks of the sputtered atoms.
The tracks of the sputtered particle in the sputtering apparatus are calculated by applying the MC method to each of the ejection angle data. In this calculation, the collision of the particles through a central force of a Lennard-Jones type potential and the trapping of the sputtered atom by the collimator and the wall of the arrangement are taken into account (step S51).
Finally, of the sputtered particles that have kept the tracks thus calculated, the particles, which arrive at a specific region on a wafer, are extracted (steps S52, S53). The shape of the region where the particles are actually to arrive is then calculated in step S6 with the aid of the string model or the like.
In this calculation, steps P21 to S53 are repeated until M ejection angle data are served to the MC calculation (step S54), wherein M is the number of the ejection angle data desirable to minimize the random number error in the MC calculation. M need be of the order often millions. Here, the random number error refers to an error originating from the deviation from a result of calculation carried out on the assumption of an infinite number of random numbers.
In the foregoing method of simulation, substantial sampling errors in directional components (direction cosines with respect to x-, y- and z-axes) of the possible tracks of the sputtered particles at their ejection points depend on the number of ejection angle data N.
Since the value of N, however, is taken as small as 100-200 because of the limitation in practical calculation time, a random number error is as large as 1/Nxc2xd (≅10xe2x88x921) can occur in this case. As a result, the calculated shape of the formed film becomes deformed due to a 20 to 30% error as compared with an ideal shape (the shape assumed in the case of a negligible-random number error) of the film.
The problem to be solved by the present invention is summarized as follows:
The foregoing method of simulation needs ejection angle data of the order of ten millions to be taken into the calculation (the desirable number of the ejection angle data to be taken into calculation will be hereinafter denoted as M) in order to minimize the random number error. In addition, because the sampling error decreases as N is increased, it is advantageous to make the number of the ejection angle data N as great as possible in order to minimize the sampling error. (The sampling error refers to an error which is likely to occur when directional components of the possible tracks of the particles are sampled at their ejected points). The value of the ejection angle data N, however, needs to be kept as small as 100-200 because of the limitation in calculation time, as described above. Consequently, a problem in Yamada""s former method has been that a large random number error takes place from such a small value of N.
It is an object of the present invention to provide a method of simulating sputtering by which a random number error can be reduced and thus a film with a shape of a precise size can be formed by sputtering.
In order to realize the above object, the present invention is directed to calculating an direction-dependent distribution of ejected particles, effecting a calculation of tracks of the sputtered particles according to the MC method within a practically performable time and accuracy and obtaining an accurate shape of a produced film.
In order to attain the objects of the invention, the method of simulating sputtering according to the present invention, comprises:
a first step of calculating a direction-dependent distribution of ejected particles from the target;
a second step of dividing a range of the vertical angle xcex8 into sections of an equal interval, counting a number of the ejected particles for every section of the vertical angle xcex8, and calculating a vertical distribution function by interpolating the counted numbers of the ejected particles as a function of the vertical angle xcex8,
a third step of determining values of said vertical angle xcex8 likely to emerge in a random process of a particle ejection from the target using the vertical distribution based on the rejection method,
a fourth step of determining values of the horizontal angle xcfx86 likely to emerge in a random process of a particle ejection from said target, and
a fifth step of calculating tracks of sputtered particles in a sputtering arrangement using the values of said vertical angles and said horizontal angles determined by the third step and the fourth step in accordance with the Monte Carlo method.
The third step preferably includes:
a step of generating sets of random numbers, each set consisting of two random numbers, a first random number being for designating a value of the vertical angle xcex8, and a second random number being for comparison with a value of the vertical distribution function for a vertical angle designated by the first random number;
a step of comparing said second random number with a value of the vertical distribution function for a vertical angle designated by the first random number, and judging whether or not the second random number is equal to or less than said value of the vertical distribution function; and
a step of accepting the first random number as a vertical angle likely to emerge if the second random number is equal to or less than the value of the vertical distribution function for the designated vertical angle and rejecting the first random number if the second random number is more than the value of the vertical distribution function for the designate vertical angle.
The fourth step preferably includes a step of designating the horizontal angle by uniform random numbers when it is conditioned that a particle is ejected from said target in an equal probability for all values of said horizontal angle xcfx86 at a fixed value of xcex8.
In this case, the vertical distribution function is preferably calculated by integrating a vertical distribution function for an arbitrary given horizontal angle, with respect to xcfx86.
The method of simulating preferably further includes a step of judging the number of the vertical angles used in calculation according to the Monte Carlo (MC) method, and if the judged number of said vertical angles is less than a predetermined number M, the designations of the vertical angle and the horizontal angle by the steps 3 to 5 are repeated until the number M is attained.
The number M is determined so as to put the random number error occurring in the MC calculation in a minimum tolerable, preferably determined to be at least ten millions.
By the constituent features particularly of the second and third steps above, a large number (M) of the ejection angle data can be generated so as to adapt to the distribution function. The large number of the ejection angle data allow the MC calculation with a minimum random-number error. The distribution function can be obtained by means of the interpolation method from a comparatively small number of the ejection angle data that can be calculated within a practically allowable time. Consequently, the present invention provides a method to generate a large number of ejection angle data, which represent likely directions of ejection of particles in the random ejection process, so as to adapt to the distribution function calculated from a small number of ejection angle data.
By virtue of the above-described constituent features, the present invention offers the advantage of significantly reducing the random number error, and allowing the precise shape of a produced film to be determined.